Not sure what you mean by 'solid cork grip'. There is a photo of the full handle in the link above if you want to look at it. It doesn't have an exposed section in the rear grip like the H and XH if that is what you are asking.

Could you throw a huddleston on it, yeah you could. Is it an ideal rod for that lure, not in my opinion. I would use the H or XH for that lure.